Infi-2 8

Improper integrals

\begin{matrix} \int_{a}^{\infty} f (x) d x = ? ? ? \\ \int_{- \infty}^{b} f (x) d x = ? ? ? \\ \int_{- \infty}^{\infty} f (x) d x = ? ? ? \end{matrix}

Improper integral of the first type #definition

\begin{matrix} Function f is called Riemann-integrable on [a, \infty) iff \\ \forall b > a : f is Riemann-integrable on [a, b] \\ If function f is Riemann-integrable on [a, \infty) \\ It’s integral is called improper and is equal to \\ \int_{a}^{\infty} f (x) d x = lim_{b \to \infty} \int_{a}^{b} f (x) d x \\ Similarly, \int_{- \infty}^{a} f (x) d x = lim_{b \to - \infty} \int_{b}^{a} f (x) d x = L \\ If L \in R, improper integral is said to converge \end{matrix}

\begin{matrix} \int_{2}^{\infty} \frac{1}{x \ln x} d x = lim_{b \to \infty} \int_{2}^{b} \frac{1}{x \ln x} d x \\ Let t = \ln x ⟹ d t = \frac{d x}{x} \\ ⟹ \int \frac{1}{x \ln x} d x = \int \frac{1}{t} d t = \ln | t | = \ln | \ln x | + C \\ ⟹ lim_{b \to \infty} \int_{2}^{b} \frac{1}{x \ln x} d x = lim_{b \to \infty} \ln | \ln b | - \ln | \ln 2 | = lim_{b \to \infty} \ln (\ln b) = \infty \\ ⟹ This integral diverges \end{matrix}

\begin{matrix} \sum_{n = 2}^{\infty} \frac{1}{n \ln (n)} also diverges, is there a connection? \end{matrix}

\begin{matrix} \int_{0}^{\infty} x e^{- x} d x \\ f (x) = x, g^{'} (x) = e^{- x} \\ f^{'} (x) = 1, g (x) = - e^{- x} \\ \int x e^{- x} d x = - x e^{- x} + \int e^{- x} d x = - x e^{- x} - e^{- x} + C \\ ⟹ \int_{0}^{\infty} x e^{- x} d x = lim_{b \to \infty} \int_{0}^{b} x e^{- x} d x = lim_{b \to \infty} (- b e^{- b} - e^{- b} + e^{- 0}) = lim_{b \to \infty} \frac{- b}{e^{b}} + 1 = \\ = - 0 + 1 = 1 \\ ⟹ This integral converges to 1 \\ Note: \forall n \in N_{0} : \int_{0}^{\infty} x^{n} e^{- x} d x = n! \\ Proof: \\ Let Γ (n) = \int x^{n} e^{- x} d x \\ Γ (n + 1) = \int x^{n + 1} e^{- x} d x = - x^{n + 1} e^{- x} + (n + 1) \int x^{n} e^{- x} d x = \\ = - x^{n + 1} e^{- x} + (n + 1) Γ (n) \\ Γ (0) = - e^{- x} \\ ⟹ Γ (n) = \sum_{i = 0}^{n} \frac{n!}{(n - i)!} x^{n - i} (- e^{- x}) \\ ⟹ \int_{0}^{\infty} x^{n} e^{- x} d x = lim_{b \to \infty} Γ (n) |_{x = 0}^{x = b} = \\ = lim_{b \to \infty} \sum_{i = 0}^{n} \underset{\to 0}{\underset{⏟}{\frac{(n)!}{(n - i)!} b^{n - i} (- e^{- b})}} - n! (- e^{- 0}) = n! \end{matrix}

Improper integral from -inf to +inf #definition

\begin{matrix} \int_{- \infty}^{\infty} f (x) d x = \int_{- \infty}^{a} f (x) d x + \int_{a}^{\infty} f (x) d x \\ \int_{- \infty}^{\infty} f (x) d x converges ⟺ Both \int_{- \infty}^{a} f (x) d x, \int_{a}^{\infty} f (x) d x converge \end{matrix}

\begin{matrix} Note: \int_{- \infty}^{\infty} f (x) d x \neq lim_{b \to \infty} \int_{- b}^{b} f (x) d x \\ Example: \\ \int_{- \infty}^{\infty} x^{7} d x = \int_{- \infty}^{a} x^{7} d x + lim_{b \to \infty} \int_{a}^{b} x^{7} d x = \int_{- \infty}^{a} x^{7} d x + \underset{\infty}{\underset{⏟}{lim_{b \to \infty} (\frac{b^{8}}{8} - \frac{a^{8}}{8})}} = \infty \\ lim_{b \to \infty} \int_{- b}^{b} x^{7} d x = lim_{b \to \infty} (\frac{b^{8}}{8} - \frac{(- b)^{8}}{8}) = 0 \end{matrix}

Convergence tests

p-Integral test for improper integrals of the first type #lemma

\begin{matrix} \int_{a}^{\infty} \frac{1}{x^{p}} d x converges ⟺ p > 1 \\ Proof: \\ \int_{a}^{\infty} \frac{1}{x^{p}} d x = lim_{b \to \infty} \int_{a}^{b} \frac{1}{x^{p}} d x \\ Let p = 1 \\ \int_{a}^{b} \frac{1}{x^{p}} d x = \ln | b | - \ln | a | ⟹ \int_{a}^{\infty} \frac{1}{x^{p}} d x = lim_{b \to \infty} (\ln | b | - \ln | a |) = \infty \\ Let p \neq 1 \\ \int_{a}^{b} \frac{1}{x^{p}} d x = \int_{a}^{b} x^{- p} d x = \frac{b^{1 - p}}{1 - p} - \frac{a^{1 - p}}{1 - p} \\ ⟹ lim_{b \to \infty} \int_{a}^{b} \frac{1}{x^{p}} d x = lim_{b \to \infty} \frac{1}{1 - p} (b^{1 - p} - a^{1 - p}) \\ p < 1 ⟺ b^{1 - p} \to \infty ⟺ \int_{a}^{\infty} \frac{1}{x^{p}} d x diverges \\ p > 1 ⟺ b^{1 - p} \to 0 ⟺ \int_{a}^{\infty} \frac{1}{x^{p}} d x converges \end{matrix}

Comparison test for improper integrals of the first type #lemma

\begin{matrix} Let f, g be Riemann-integrable on [a, \infty) \\ Let 0 \leq f \leq g \\ Then \int_{a}^{\infty} g (x) d x converges ⟹ \int_{a}^{\infty} f (x) d x converges \end{matrix}

Limit comparison text for improper integrals of the first type #lemma

\begin{matrix} Let f, g be Riemann-integrable on [a, \infty) \\ Let 0 \leq f, g \\ L = lim_{x \to \infty} \frac{f (x)}{g (x)} \\ \begin{matrix} L = \infty ⟹ & [\int_{a}^{\infty} f (x) d x converges ⟹ \int_{a}^{\infty} g (x) d x converges] \\ L = 0 ⟹ & [\int_{a}^{\infty} f (x) d x converges ⟸ \int_{a}^{\infty} g (x) d x converges] \\ 0 < L < \infty ⟹ & [\int_{a}^{\infty} f (x) d x converges ⟺ \int_{a}^{\infty} g (x) d x converges] \end{matrix} \end{matrix}